First just to say I'm directly answering the question - why might some Tibetans not want to be part of China. Whether it is practical or possible for Tibet to be a separate country is another matter altogether, as it is often that people want things they can't have.
Tibetan culture is very different from Chinese. For one thing, their scripts are mutually unintelligible.
The Chinese all use Chinese pictograms, which mean that a Chinese in any part of China can understand any other Chinese person.
Tibetans use a completely different, phonetic script related to Indian scripts
A Tibetan can't read Chinese and vice versa.
Also the language is in a different branch, the Tibetic languages rather than the Chinese languages, Sino-Tibetan languages (though Chinese also speak many mutually unintelligible languages).
There the script makes a big difference, Chinese who can't understand each other's languages can still communicate using the script, because it is pictographic, communicating concepts rather than phonetics.
Actually, the common written language which unified the Chinese empire was a dead language, like our Latin. How did the Chinese manage to be unified by a common written language given the huge size and population of China when Europe with a smaller size and population is divided by different languages?
They mention the Shi poem which is intelligible in the classical written language but not in spoken Mandarin (though apparently much easier to understand in some dialects with more vowel sounds than Mandarin Lion-Eating Poet in the Stone Den)
Anyway even though it’s a dead language, it did unify China and they all use the same pictogram type script even though modern versions of Chinese are often not mutually intelligible any more (thanks for the correction, Quora User)
But Tibetans don't use this script, instead using their phonetic script which can only convey meaning to other Tibetans.
Then as for the religion, Tibetan Buddhism is a different branch of Buddhism from Chinese Buddhism. In both cases, introduced directly from India. And has developed differently.
Many practices in Tibet are more closely related to some Indian customs, e.g. the practice of sky burial and charnel grounds which relate to customs of some minority Indian religions - and in ancient times were common in India.
Tibetan Buddhism has a long established system of recognizing incarnations of some of its spiritual teachers when they die, identified as a young child of a few years old through various signs. Chinese Buddhism doesn't have this tradition.
The culture is very different.
So, if it is natural for many Scottish people, to want to be independent of the UK, as in our recent referendum, even when we all speak the same language and belong to the same small island and have a largely shared culture - of course it is natural for many Tibetans to want to be independent from China.
In the past at times China has invaded Tibet and taken control of it, and other times it has left it alone. This is the most recent of several such invasions.
Some Tibetans call for Tibet to be free from China. That's the Free Tibet movement.
The Dalai Lama calls for a more moderate position - that Tibet remains part of China but with greater autonomy. He is asking mainly for religious, educational, and cultural freedom and freedom to make their own decisions to deal with environmental issues in Tibet.
Whatever happens, perhaps it can help for all those concerned to recognize that it is natural for this wish to arise in many Tibetans just as it has arisen in many Scottish people in the UK. There is nothing wrong with the wish, it is totally natural. But what one does about it, if anything, is another thing altogether.
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As a Buddhist in one of the Tibetan traditions my sympathies are with the second, autonomous region as suggested by the Dalai Lama, because it seems the most practical one with at least a tiny chance that it might actually happen, while preserving freedom for many of the things Tibetans care most about.
As someone living in Scotland, who recently voted for Scotland to become an independent country in the referendum, then my sympathies are with the first choice, so long as it was achieved through peaceful means. If I was Tibetan living in Tibet, I'm pretty sure I'd want it to be a separate country.
But even if that was ones wish, it is probably not achievable, while the middle possibility - just possibly might be. I hope that some day the Tibetans get more autonomy to govern things that can be of no danger or harm to the Chinese, indeed a benefit surely to let those on the spot make the decisions - and allowed to get on with their lives as they like to lead them themselves, and follow their own religious ideas and so forth.
Of course any transition like that would need to be done gradually. If Tibet was made autonomous overnight, then - there probably wouldn't be the infrastructure in place to make it work properly.